The Kinematics of Conical Involute Gear Hobbing

The Kinematics of Conical Involute Gear Hobbing Carlo Innocenti (This article fi rst appeared in the proceedings of IMECE2007, The technical literature contains plenty of information the 2007 ASME International Mechanical Engineering regarding the tooth fl ank geometry (Refs. 8–9) and the setting Congress and Exposition, November 11–15, 2007, Seattle, of a hobbing machine in order to generate a beveloid gear Washington, USA. It is reprinted here with permission.) (Refs. 10–15). Unfortunately, the formalism usually adopted makes determination of the hobbing parameters a rather involved process, mainly because the geometry of a beveloid Management Summary gear is customarily—though ineffi ciently—specifi ed by It is the intent of this presentation to determine all rigid-body positions of two conical involutes that resorting to the relative placement of the gear with respect mesh together, with no backlash. That information to the standard rack cutter that would generate the gear, even then serves to provide a simple, general approach in if the gear is to be generated by a different cutting tool. To arriving at two key setting parameters for a hobbing make things worse, some of the cited papers on beveloid gear machine when cutting a conical (beveloid) gear. hobbing are hard reading due to printing errors in formulae A numerical example will show application of the and fi gures. presented results in a case study scenario. Conical This paper presents an original method to compute the involute gears are commonly seen in gearboxes parameters that defi ne the relative movement of a hob with for medium-size marine applications—onboard respect to the beveloid gear being generated. Pivotal to the engines with horizontal crankshafts and slightly proposed method, together with a straightforward description sloped propeller axes—and in automatic packaging of a beveloid gear in terms of its basic geometric features, is applications to connect shafts with concurrent axes the determination of the set or relative rigid-body positions whenever the angle between these axes is very small of two tightly meshing beveloid gears. Based on this set of (a few degrees). relative positions, the paper shows how to assess the rate of change of the hob-work shaft axis distance as the hob is fed Introduction across the work, as well as the rate of the additional rotation Conical involute gears, also known as beveloid gears, of the hob relative to the gear. These parameters have to be are generalized involute gears that have the two fl anks of the entered into the controller of a CNC hobbing machine in order same tooth characterized by different base cylinder radii and to generate a given beveloid gear. different base helix angles (Refs. 1– 4). Beveloid gears can be The proposed method also can be applied to the grinding of mounted on parallel-, intersecting- or skew-axis shafts. They beveloid gears by the continuous generating grinding process. can be cheaply manufactured by resorting to the same cutting Furthermore, it can be extended to encompass the cases of machines and tools employed to generate conventional, involute swivel angle modifi cation and hob shifting during hobbing. helical gears. The only critical aspect of a beveloid gear pair, Lines of Contact i.e., the theoretical punctiform (single-point) contact between The hob that generates a beveloid gear in a hobbing the fl anks of meshing gears, can be offset by a careful choice machining process can be considered an involute gear. Most of the geometric parameters of a gear pair (Refs. 5–7). On the commonly, such a gear is of the cylindrical type, although other hand, the localized contact between beveloid gear teeth adoption of a conical involute hob would be possible too, in comes in handy should the shaft axes be subject to a moderate, principle. For this reason, the kinematics of hobbing will be relative position change in assembly or operation. presented in this paper by referring to a beveloid hob, and 0056 GEARTECHNOLOGY July 2008 www.geartechnology.com subsequently specialized to the case of a cylindrical hob. This section introduces the nomenclature adopted in the (4) paper and summarizes known results pertaining to the loci of contact of a conical involute gear set. The reader is referred Owing to Equation 4, the following equations and ensuing to References 16 and 17 for a detailed explanation of the text and fi gures will refer to the normal base pitch of right-hand reported concepts and formulas. and left-hand fl anks of both gears as p1 and p–1 respectively. The tooth fl anks of a conical involute gear are portions The common perpendicular to the axes of a pair of of involute helicoids. As opposed to classical helical involute meshing beveloid gears intersects the axes themselves at gears, the two helicoids of the same tooth of a conical involute points A1 and A2 (Fig. 2). The relative position of these axes gear do not generally stem from the same base cylinder, nor continued have the same lead. In a beveloid gear set composed of gears G1 and G2, the axis of gear G1 is here directed in either way by unit vector n1. The orientation of the axis of gear G2 by unit vector n2 is then so chosen as to make the right-hand fl anks of gear G come into contact with the right-hand fl anks of gear G . 1 2 �i With reference to Figure 1, the distinction between right-hand Fi and left-hand tooth fl anks is possible based on the sign of the �i,1 ensuing quantity: �i �i,-1 �i (1) �i,1 where Fi is a point on a tooth fl ank of gear Gi (i=1, 2), Oi is a point on the axis of gear Gi, and qi is the outward-pointing unit �i vector perpendicular to the tooth fl ank at Fi. A tooth fl ank is a right-hand or left-hand fl ank, depending on whether Equation 1 results in a positive or, respectively, negative quantity. (In Figure 1, point Fi lies on a right-hand fl ank of gear Gi). In the sequel, index j will be systematically used to refer to right- Figure 1—Basic dimensions of beveloid gear Gi. hand (j = + 1) or left-hand (j = −1) tooth fl anks. Figure 1. Basic dimensions of beveloid gear G i. The basic geometry of gear Gi (i=1, 2) is defi ned by its number of teeth Ni, the radii ρi,j (j = ± 1) of its base cylinders, the base helix angles β of its involute helicoids (β < i,j i,j �� π/2; βi,j > 0 for right-handed helicoids), and the base angular y� �� thickness ϕ′i of its teeth at a specifi ed cross section. All these n parameters—with the exception of N —are reported in Figure � � i �� 1, which also shows the involute helicoids as stretching x� �� inwards up to their base cylinders, irrespective of their actual radial extent. �� The normal base pitch is the distance between homologous involute helicoids of adjacent teeth of the same gear. A � beveloid gear has two normal base pitches—pi,1 and pi,–1— one for right-hand fl anks and one for left-hand fl anks. Their expression is: �� x� �� (2) �� where y� z� �� (3) �� n� Two beveloid gears can mesh only if they have the same normal base pitches, i.e., only if the ensuing conditions are satisfi ed: Figure 2—The lines of contact. �� www.geartechnology.com July 2008 GEARTECHNOLOGY 0057 (8) P3–i, –1 P 3–i,1 (9) (10) xi (11) Pi, –1 Pi,1 The function sign(.) in Equation 10 returns the value + 1, �i,1 �i, –1 0, or –1 depending on whether its argument is positive, zero or negative. In Figure 3, angle θi,1 is positive, whereas angle θi,–1 is negative. Ai Due to the square root in Equation 6, two beveloid gears yi can properly mesh only if condition �i,1 �i, –1 (12) is satisfi ed for both values of j. In the following equations, text and fi gures Equation 12 will be supposed as holding. Figure 3—Coordinates of the extremities of the lines of contact. Thanks to Equations 5 and 6, angle θ can be expressed Figure 3 . Coordinates of the extremities of the lines of contact. i,j as: is defi ned by their mutual distance a0, together with their relative inclination α . Specifi cally, angle α is the amplitude 0 0 (13) of the virtual rotation—positive if counterclockwise—about vector (A2–A1) that would make unit vector n1 align with unit vector n . Two fi xed Cartesian reference frames W (i =1, 2) 2 i (i=1,2; h= 3 – i; j= ± 1). are then introduced with origins at points A , x axis pointing i i The z-coordinate b of point P in reference frame W towards A , and z axis directed as unit vector n . i,j i,j i 3– i i i (i=1,2; j= ± 1) is given by: For a conventional beveloid gear set composed of two b = meshing conical involute gears that revolve about their fi xed 1,j and non-parallel axes, the locus of possible points of contact (14) between the involute helicoids of right-hand (left-hand) fl anks is a line segment that has a defi nite position with respect to either of reference frames W (i=1, 2). More specifi cally, the i Based on the relative positions of reference frames W and line of contact is tangent at its ending points P and P 1 1,1 2,1 W , together with the cylindrical coordinates ρ , θ , and b of (P and P ) to the base cylinders of the right-hand (left- 2 i,j i,j i,j 1,–1 2,–1 points P with respect to W (i=1,2; j= ± 1), the length σ of the hand) fl anks of the two gears.

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